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Finding angles and area in hexagon?

In the figure ABCDEF is a regular hexagon whose side is 4. Inscribed in a rectangle PQRS

a) Find the measure of each interior angle of the hexagon

b)The diagonals meet at O(center). What is the nature of triangle AOF? Why?

C) Find the area of triangle AOF

d)What is the area of ABCDEF

e) what ia the measure of angle PAF

F) what is the area of PQRS

Thank u so much, i have an exam and i couldnt solve this, it was a suggested problem

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  • Viola
    Lv 5
    6 years ago
    Favorite Answer

    a) Exterior angle of a regular hexagon is 360/6 = 60°, so interior angle is 180 - 60 = 120°.

    b) Because the hexagon is regular, the angle AOB must e 360/60 = 60°. Also by regularity, AO = BO, so AOB is isosceles. Therefore the other two angles are also 60°, so the triangle is equilateral.

    c) Area of triangle is half base times height. Taking the base as 4 (units not given?), then imagine chopping such a triangle down the middle, making a right-angled triangle whose hypotenuse is 4 (one of the sloping sides), whose base is 2 (half the eq. triangle base), and whose perpendicular height is therefore, by Pythagoras' theorem, the square root of 4^2 - 2*2, which is 2 sqrt 3.

    So the area of the eq. triangle is 0.5 * 4 * 2 sqrt 3, or 4 sqrt 3, approx. 6.928.

    d) Since the hexagon is composed of six equal such eq. triangles, its area must be 24 sqrt 3, approx. 41.57.

    e) PAB is a straight line, FAB is one of the interior angles of the hexagon therefore 120°, so PAF must be 180 - 120 or 60°.

    f) PAF is one of those right-angled triangles described in (c). Therefore PA = 2 and PF = 2 sqrt 3. So the base of the rectangle in question is 2 + 4 + 2 = 8, and its height is 2 * 2 sqrt 3 = 4 * sqrt 3.

    Therefore its area is the product of these numbers, i.e. 32 * sqrt 3, or approx. 55.43.

    • Kitty6 years agoReport

      Thank you so so much!! You have enlightened me!

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  • 6 years ago

    I created a formula to calculate the interior angle of a regular hexagon

    The formula is Angle = 180degrees - (360degress)/n where n is the number of sides....

    Here if it has 4 sides the inteior angle is 180 - 360/4 = 180 - 90 =

    90 degrees. OK! To confirm just graph the regular hexagon with 4 sides... it is a square... then the interior angle has 90 degrees OK!

    • Kitty6 years agoReport

      Really thank you so so so much!!!

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